How to avoid pink in EEVEE
In addition to my comment above, I did a little research regarding what to expect in the future. It seems that it's not (only) a limitation in Eevee, but also depends heavily on the GPU/driver.
The Blender documentation states the following:

Memory Management
In Eevee, GPU Memory management is done by the GPU driver. In theory, only the needed textures and meshes (now referred as “the resources”) for a single draw call (i.e. one object) needs to fit into the GPU memory.
So if the scene is really heavy, the driver will swap things in and out to make sure all objects are rendered correctly.
In practice, using too much GPU memory can make the GPU driver crash, freeze, or kill the application. So be careful of what you ask.
There is no standard way of estimating if the resources will fit into the GPU memory and/or if the GPU will render them successfully.

Since Eevee is a real-time rendering engine (much like any other graphics engine found in modern games) based on OpenGL, it inherits all the limitations that come with OpenGL, the driver and the GPU.
The GPU drivers are designed to work as fast as possible, so they don't bother with sending messages back to the application to inform about a limit being reached.
Therefore it seems highly unlikely that we see any improvement of the current limitations in the near future.